# Probability of an event in Merged Poisson Process

I have two Poisson processes A and B having arrival rates $$\lambda_{1}$$ and $$\lambda_{2}$$.

What is the probability that only 1 arrival appears from Process A and 0 arrival occurs from process B in the time interval [0-10s]? I mean what is the probability that in 10s time only 1 arrival from Process A appears?

Well, $~\mathsf P(A_{(0;10]}{=}1 \cap B_{(0;10]}{=}0) ~=~ \mathsf P(A_{(0;10]}{=}1)\,\mathsf P(B_{(0;10]}{=}0)~$ if they are independent processes, which you neglected to mention if it is so or not.